Not a whole lot of residential areas in the Kenai, you'll see, plus who wants to be by them anyway?
Good thing is plenty of NF, BLM, State, County-City and Wilderness areas to camp. Best to always ask if unsure.
Places like Centennial Campground, https://soldotna.org/departments/parks-and-recreation/parks-amenities/centennial-park just outside of Soldotna are a trick to see-find but are around, ask locals when in town. It's a nice place to camp if the Salmon are running, big Kings, plus you can walk into town. It is close to town so you can get a few losers/drunks at time around the campground,no big deal just be aware. Don't forget about Anchor Point.
Avoid any private land, some of those folks like it off the grid for a reason.............
IMO, both Fairbanks and Anchorage are boring and overall ugly places, no real reason to stay. Get your rental and supplies and goooooo.
Don't know when you land but consider getting out of Anchorage straight away, hitchhike or train down to Girdwood, way more pleasant and in your in nature. Plus your right on Turnagain Arm and might get lucky and see a good size Bore Tide, really cool to see.
The entire state is amazing but 2 places not mentioned much are the Haul Road to at least Coldfoot, amazing remote killer scenic country and McCarthy-Wrangell St. Elias NP and the road to the tiny town of McCarthy, fantastic scenery and tons of wildlife.
If you decide make sure your vehicle is allowed on gravel roads, they are available. Also might check into 4x4 truck camper rentals, good way to explore the state. Transport and lodging in one, plus you can get off road...... about the same cost as a car rental and regular lodging.
By the way, if you head down to Seward about 6 miles south of Moose Pass off Hwy.9 is Trail River Campground, about a half hour before Seward. Nice little campground on Kenai Lake, fun hiking along the lake, few people. Camp away from the lake if you do, too cold by it, even in summer.
